Armodue harmony: Difference between revisions
Wikispaces>hstraub **Imported revision 171656715 - Original comment: ** |
Wikispaces>hstraub **Imported revision 171657239 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-10-19 02: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-10-19 02:59:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>171657239</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 27: | Line 27: | ||
In Armodue (see [[16edo]], in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness. | In Armodue (see [[16edo]], in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness. | ||
For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[seventh | For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7_4|seventh]] harmonic. | ||
==The triple mean of the double diagonal / side of the square== | ==The triple mean of the double diagonal / side of the square== | ||
| Line 59: | Line 59: | ||
The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents. | The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents. | ||
This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism | This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour. | ||
In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system. | In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system. | ||
| Line 65: | Line 65: | ||
==The intervals of 2 eka and 14 eka== | ==The intervals of 2 eka and 14 eka== | ||
The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic | The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic of the overtone series. It is the interval that is obtained by dividing the tenth of Armodue (the tempered octave) into eight equal parts, to form the [[8edo|8-equal tempered]] scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures. | ||
In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka ( 150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence "speculative harmony". In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more of notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context. | In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka ( 150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence "speculative harmony". In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more of notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context. | ||
| Line 389: | Line 389: | ||
In Armodue (see <a class="wiki_link" href="/16edo">16edo</a>, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.<br /> | In Armodue (see <a class="wiki_link" href="/16edo">16edo</a>, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.<br /> | ||
<br /> | <br /> | ||
For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the <a class="wiki_link" href="/ | For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the <a class="wiki_link" href="/7_4">seventh</a> harmonic.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Two theses supporting the system-The triple mean of the double diagonal / side of the square"></a><!-- ws:end:WikiTextHeadingRule:4 -->The triple mean of the double diagonal / side of the square</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Two theses supporting the system-The triple mean of the double diagonal / side of the square"></a><!-- ws:end:WikiTextHeadingRule:4 -->The triple mean of the double diagonal / side of the square</h2> | ||
| Line 421: | Line 421: | ||
The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.<br /> | The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.<br /> | ||
<br /> | <br /> | ||
This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism | This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.<br /> | ||
<br /> | <br /> | ||
In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.<br /> | In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.<br /> | ||
| Line 427: | Line 427: | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="The interval table-The intervals of 2 eka and 14 eka"></a><!-- ws:end:WikiTextHeadingRule:12 -->The intervals of 2 eka and 14 eka</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="The interval table-The intervals of 2 eka and 14 eka"></a><!-- ws:end:WikiTextHeadingRule:12 -->The intervals of 2 eka and 14 eka</h2> | ||
<br /> | <br /> | ||
The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic | The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic of the overtone series. It is the interval that is obtained by dividing the tenth of Armodue (the tempered octave) into eight equal parts, to form the <a class="wiki_link" href="/8edo">8-equal tempered</a> scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.<br /> | ||
<br /> | <br /> | ||
In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka ( 150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence &quot;speculative harmony&quot;. In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more of notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context.<br /> | In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka ( 150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence &quot;speculative harmony&quot;. In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more of notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context.<br /> | ||